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I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.

I'm hoping some Algebraist could help me properly state the following:

A number in and of itself has no true meaning. True in the sense that it relates to an existing object within our world. The question we need to ask is how do we teach children meaning if that meaning is not at first grounded within something concrete.

Numbers in and of themselves represent abstract notions and in the pure study of mathematics we study mostly patterns: the various patterns that emerge from these abstract notions and the various means through which some relation is developed or expressed between them. Meaning that between the value 1 and 2 for instance there is no relation except when explicitly defined for example as some additive operation, in general an addition of multiples of the unit element.

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    $\begingroup$ Could you give an example of a confusion children have that you want to help with by explaining to teachers about number? I am not a primary teacher, but it is my experience that children tend to do well with the concreteness of number, i.e., one-ness and two-ness properties of abstract classes of objects, and it is operations on the numbers where things become tricky for them. This is maybe what you are getting at with "we study mostly patterns"? $\endgroup$
    – Carser
    Jan 6 at 12:52
  • $\begingroup$ @Carser So a number is an abstract concept. We have to attach some meaning/context to it (when teaching early years, something Montessori education does well in). Regarding Algebra (Abstract Algebra?) we are mostly engaged in some form of pattern recognition studying/identifying the underlying structure/relations. $\endgroup$
    – Johnny M
    Jan 6 at 13:43
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    $\begingroup$ This sounds a lot more like the confusions faced by undergraduate philosophy majors. I don't disagree with anything that you say, but the plain truth is that arithmetic and geometry served humanity exceptionally well for thousands of years before we had any useful progress on understanding what a number was. $\endgroup$ Jan 6 at 13:44
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    $\begingroup$ I'm sorry, but this just sounds misguided. Even if you could clearly formulate some abstract philosophical point along these lines, a prerequisite for understanding your abstract idea would be that your audience would need a firm grasp of the subject's concrete aspects. If only we lived in a world where all preservice K-6 teachers had this level of fluency. In reality, in my experience, most of them are afraid of math, and many lack a basic working understanding of things like the meaning of multiplication, or how to tell whether 1/3 is greater than 1/2. $\endgroup$
    – user507
    Jan 6 at 18:39
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    $\begingroup$ I agree with @Carser (and the comments by Daly and Crowell), and I think this is a solution in search of a problem. Also, I think if you dispense with the heavy-handed language and discuss the distinction between concrete numbers (e.g. $2$ rocks) and abstract numbers (e.g. $2),$ you'll find that this was probably covered in math-education and child development teacher education courses. Of course, "covered" and "understood and retained" are two different things . . . $\endgroup$ Jan 6 at 21:33
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I highly recommend the book The Number Sense: How the Mind Creates Mathematics, by Stanislas Dehaene and published by Oxford University Press. Another book that comes to mind is The Language Instinct: How The Mind Creates Language, by Steven Pinker. Both books have had broad impact on scholarship related to your question.

As you try to refine your question and find an answer that satisfies you, it might help to think about how artificial intelligence "learns." Let's take the sentence from your OP "The question we need to ask is how do we teach children meaning if that meaning is not at first grounded within something concrete." and replace "children" by "machines" or "artificial intelligence." To be sure, there is much controversy about all of this, related to meaning and understanding.

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  • $\begingroup$ I might add "Number: the language of science" by Dantzig as well as "Arithmetic" by Lockhart. $\endgroup$
    – Carser
    Jan 6 at 17:41
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I would suggest taking it easy to start but maybe something like base numeration is a way in here. There is no reason we have to use base 10 and students use of different bases can be very important to understanding operations that require regroupings, and that weird alignment under the typical presentation of multiplication algorithms.

One example to check out is Roger Howe's essay here. I found activities where teachers count, add, subtract, and multiply using base 4 blocks to be very effective in my elementary mathematics teaching courses.

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  • $\begingroup$ Base 4 because four limbs? $\endgroup$
    – Rusty Core
    Jan 7 at 22:26
  • $\begingroup$ I like where you’re going with this Rusty... $\endgroup$
    – jfkoehler
    Jan 7 at 22:45
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Forget New Math, it does now work with young kids, and it is not needed for learning arithmetics in elementary school.

Instead, first teach the concepts of "same as", "more than", "less than" by lining up real objects like apples or backpacks or horses one against another, making pairs. Then you ask, how do we figure out "more" or "less" or "same" without bringing my horses and your horses to the river? We can abstract horses into counters and line up counters mine against yours and compare them. Then you ask, can we abstract other things like cows or women or children with counters? Do they have to be the same counters? Can we use different ones? Can we use sticks instead of stones? Can we compare without lining up counters? Well, we write counters on paper instead of actually carrying stones or sticks. Pictures of counters is an abstraction of physical counters. When we get too many things, it becomes unwieldy, can we do better? Well, we can "count" the counters - a whole new idea and a process, and come up with - ta-da! - numbers. Comparing 3 and 5 is the same as comparing ||| and |||||, just different representation. How is it better? Without introducing place value, not much. So you explain how place value works, and how with limited number of digits we can now represent any countable number, another level of abstraction.

But please, don't tell kids that numbers don't exist, that they represent the abstract notion blah-blah-blah. They tried it sixty years ago, and it did not work.

Let me quote Why Johnny Can't Add by Morris Kline:

"Is 7 a number," asks a teacher. The students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedience causes them to reply affirmatively. The teacher is aghast. "If I asked you who you are, what would you say?"

The students are now wary of replying, but one more courageous youngster does do so: "I am Robert Smith."

The teacher looks incredulous and says chidingly, "You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? 0f course not! It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number.

The teacher sees that the students do not appreciate the distinction and so she tries another tack. "Is the number 3 half of the number 8?" she asks. Then she answers her own question: "Of course not! But the numeral 3 is half of the numeral 8, the right half."

The students are now bursting to ask, "What then is a number?" However, they are so discouraged by the wrong answers they have given that they no longer have the heart to voice the question. This is extremely fortunate for the teacher, because to explain what a number really is would be beyond her capacity and certainly beyond the capacity of the students to understand it. And so thereafter the students are careful to say that 7 is a numeral, not a number. Just what a number is they never find out.

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    $\begingroup$ Note that the question asks about defining these concepts for the teachers, not the children. $\endgroup$
    – Tommi
    Jan 7 at 15:13
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    $\begingroup$ You should define "New Math" and give context. My elementary school New Math workbook from 1965 has picture of boys, girls, jacks, dolls, and toy soldiers lined up to explain equality and inequality. In the workbook, we drew matching lines between dolls and girls, for example, to compare cardinality of the sets. (In retrospect, the workbook is embarrassingly sexist.) And I recall my teacher had bags of clothes pins so her students could have tactile experience. $\endgroup$
    – user52817
    Jan 7 at 17:07
  • $\begingroup$ @user52817: I have a similar workbook from my 1965-66 first grade year, although I don't recall anything about equality and inequality, only matching objects by drawing lines. Maybe you mean the situation when something on the left-hand side remains unmatched or something on the right-hand side remains unmatched. I still have the book (paperback workbook), and if I get the chance in the next few days (and remember to do so), I'll look around for it and give its bibliographic information. I have an idea of some places where it might be, but I don't have time now to look for it. $\endgroup$ Jan 7 at 18:18
  • $\begingroup$ @Tommi: most elementary school teachers are no better in their grasp of math than their students, so no difference, really. $\endgroup$
    – Rusty Core
    Jan 7 at 22:12
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    $\begingroup$ A reference to the teachers being no better than the pupils would be nice, then, and an argument from there that the teachers are no more philosophically mature than their pupils. Or some other argument connecting your answer to the question that was asked. $\endgroup$
    – Tommi
    Jan 8 at 7:51
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I teach kids, small one 5-6 years to 11-12 years. So I have realised that a kid's life itself is not bound to anything concrete. They are best receptacles of Knowledge. But they are not good at reproducing it. See kids are capable of learning anything however abstract, but the trick is that teacher should be visualising it within. As long as it is "real" in the teacher's imagination, kids will learn it. If it is obscure, abstract, they will not be able to describe what they saw/learned but they learned. They do look into your soul. Personal experience.

As far as numbers is concerned I don't know whether this is the answer but this is what I did. I realised that there is a path to Knowledge and one can choose any, but then there is "THE KNOWLEDGE" ITSELF without ANYTHING. So the usual path is from "things" to "Numbers". "What is this?", "An apple Teacher!", "No, it's ONE APPLE!". I thought I'll take kids from Numbers to things. So I had to Explain NUMBER ITSELF without any "thing". I told them that a number is nothing but a series of empty places, each place has a name(One's, Two's) etc. and each place has a value(1, 10) etc. So what does each place contains? A NUMBER. << Right here I explain to them difference between number and numerals. So each place contains ONLY ONE numeral. So Numeral 2 at 10's place means that this Number contains 2 tens in it. When this series of numerals is attached to a "thing" or property of a thing, then it is known as a NUMBER. So Numerals tell about a "number" and a "number" tells us about things.

And thankfully kids get it...

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